Chapter 8, Problem 32QP

Summary Introduction

To determine: The dividend yield for each four stocks.

Introduction:

Dividend is a sum of money paid to the shareholders of the company. It is distributed among the investors from a portion of company’s earnings. This can be issued or paid as shares of stock or cash payment.

Dividend yield is a ratio that specifies how much a company pays as dividends every year on comparing with its share price. It is considered as the return on investment for a stock.

Answer to Problem 32QP

The dividend yield of Stock W is 5%.

The dividend yield of Stock X is 15%.

The dividend yield of Stock Y is 20%.

The dividend yield of Stock Z is 8.8%.

Explanation of Solution

Given information:

Four different stocks have a required rate of return of 15% and the recent dividend is $3.75 per share.

The constant growth rate in dividends of Stock W is 10%.

The constant growth rate in dividends of Stock X is 0%.

The constant growth rate in dividends of Stock Y is -5%.

The constant growth rate in dividends of Stock Z is 20%.

Steps to determine the dividend yields and capital gains for each of the stocks:

• Firstly, determine the stock price for each stock. It is because all the stocks have a required return of 15%, which is the sum of dividend yield and capital gains yield.
• After determining the stock price of each stock, use the stock price and dividend to compute the dividend yield of the four stocks.
• Finally, determine the capital gains yield for the each stock by subtracting the dividend yield from the total return.

Formulae:

The formula to calculate the price of stock:

${\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}$

Where,

P_o refers to the present value of a share of stock,

D_o refers to the current year dividend paid,

R refers to the discount rate,

g refers to the constant growth of dividends.

The formula to calculate the dividend yield:

$\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}$

Where,

D₁ refers to the next period dividend per share,

P_o refers to the present value of a share of stock.

Compute the stock price of Stock W:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$3.75\times \frac{\left(1+\frac{10}{100}\right)}{\left(\frac{15}{100}-\frac{10}{100}\right)}\\ =\$3.75\times \frac{\left(1.10\right)}{\left(0.15-0.10\right)}\\ =\$3.75\times \left(\frac{1.10}{0.05}\right)\\ =\$3.75\times 22\\ =\$82.50\end{array}$

Hence, the stock price of the Stock W is $82.50.

Compute the dividend yield of the Stock W:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$3.75\times \left(1+\frac{10}{100}\right)}{\$82.50}\\ =\frac{\left(\$3.75\times 1.10\right)}{\$82.50}\\ =\frac{\$4.125}{\$82.50}\\ =0.05\end{array}$

Hence, the dividend yield of the Stock W is 0.05 or 5%.

Compute the stock price of Stock X:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$3.75\times \frac{\left(1+\frac{0}{100}\right)}{\left(\frac{15}{100}-\frac{0}{100}\right)}\\ =\$3.75\times \frac{1}{\left(0.15-0.00\right)}\\ =\$3.75\times \left(\frac{1}{\text{0}\text{.15}}\right)\\ =\$3.75\times 6.67\\ =\$25.00\end{array}$

Hence, the stock price of the Stock X is $25.00.

Compute the dividend yield of the Stock X:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$3.75}{\$25.00}\\ =0.15\end{array}$

Hence, the dividend yield of the Stock X is 0.15 or 15%.

Compute the stock price of Stock Y:

$\begin{array}{c}{\text{P}}_o={\text{D}}_o\times \frac{\left(1+\text{g}\right)}{\left(\text{R}-\text{g}\right)}\\ =\$3.75\times \frac{\left(1-\frac{5}{100}\right)}{\left(\frac{15}{100}+\frac{5}{100}\right)}\\ =\$3.75\times \frac{0.95}{\left(0.15+0.05\right)}\\ =\$3.75\times \left(\frac{0.95}{0.2}\right)\\ =\$3.75\times 4.75\\ =\$17.81\end{array}$

Hence, the stock price of the Stock Y is $17.81.

Compute the dividend yield of the Stock Y:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$3.75\times \left(1+\left(\frac{-5}{100}\right)\right)}{\$17.81}\\ =\frac{\left(\$3.75\times 0.95\right)}{\$17.81}\\ =\frac{\$3.5625}{\$17.81}\\ =0.20\end{array}$

Hence, the dividend yield of the Stock Y is 0.20 or 20%.

Compute the stock price in Year 2 of Stock Z:

$\begin{array}{c}{\text{P}}_2={\text{D}}_o\frac{{\left(1+\text{g}\right)}^2\left(1+{\text{g}}_2\right)}{\left(\text{R}-\text{g}\right)}\\ =\$3.75\times \frac{{\left(1+\frac{20}{100}\right)}^2\times \left(1+\frac{5}{100}\right)}{\left(\frac{15}{100}-\frac{5}{100}\right)}\\ =\$3.75\times \left(\frac{{\left(1+0.20\right)}^2\times \left(1+0.05\right)}{\left(0.15-0.05\right)}\right)\\ =\$3.75\times \frac{\left(1.44\times 1.05\right)}{0.1}\\ =\$3.75\times \frac{1.512}{0.1}\\ =\$3.75\times 15.12\\ =\$56.7\end{array}$

Hence, the stock price in Year 2 of Stock Z is $56.7.

Compute the current stock price of Stock Z:

$\begin{array}{c}{\text{P}}_o=\frac{{\text{D}}_1}{{\left(\text{1+R}\right)}^1}+\frac{{\text{D}}_2}{{\left(\text{1+R}\right)}^2}+\frac{{\text{P}}_2}{{\left(\text{1+R}\right)}^2}\\ =\frac{\$3.75\times \left(1+\frac{20}{100}\right)}{\left(1+\frac{15}{100}\right)}+\frac{\$3.75\times {\left(1+\frac{20}{100}\right)}^2}{{\left(1+\frac{15}{100}\right)}^2}+\frac{\$56.70}{{\left(1+\frac{15}{100}\right)}^2}\\ =\frac{\$3.75\left(1.20\right)}{1.15}+\frac{\$3.75\times {\left(1.20\right)}^2}{{\left(1.15\right)}^2}+\frac{\$56.70}{{\left(1.15\right)}^2}\\ =\frac{\$4.5}{1.15}+\frac{\left(\$3.75\times 1.44\right)}{1.3225}+\frac{\$56.70}{1.3225}\\ =\$3.91304+\frac{\$5.4}{1.3225}+\$42.87334\\ =\$3.91304+\$4.08317+\$42.87334\\ =\$50.87\end{array}$

Hence, the stock price of the Stock Z is $50.87.

Compute the dividend yield of the Stock Z:

$\begin{array}{c}\text{Dividend}\text{yield}=\frac{{\text{D}}_1}{P_o}\\ =\frac{\$3.75\times \left(1+\frac{20}{100}\right)}{\$50.87}\\ =\frac{\left(\$3.75\times 1.20\right)}{\$50.87}\\ =\frac{\$4.5}{\$50.87}\\ =0.088\end{array}$

Hence, the dividend yield of the Stock Z is 0.088 or 8.8%.

Summary Introduction

To determine: The capital gains yield of four stocks.

Introduction:

Capital gains yield is a ratio that indicates the rise in the price of common stock.

Answer to Problem 32QP

The capital gains yield of Stock W is 10%.

The capital gains yield of Stock X is 0%.

The capital gains yield of Stock Y is -5%.

The capital gains yield of Stock Z is 6.2%.

Explanation of Solution

Given information:

Four different stocks have a required rate of return of 15%. The computed dividend yield of each of the four stocks is as follows:

• The dividend yield of Stock W is 5%.
• The dividend yield of Stock X is 15%.
• The dividend yield of Stock Y is 20%.
• The dividend yield of Stock Z is 8.8%.

The formula to calculate the capital gains yield:

$\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}$

Compute the capital gains yield of Stock W:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{15}{100}-\frac{5}{100}\\ =0.15-0.05\\ =0.10\end{array}$

Hence, the capital gains yield of stock W is 0.10 or 10%.

Compute the capital gains yield of Stock X:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{15}{100}-\frac{15}{100}\\ =0.15-0.15\\ =0.00\end{array}$

Hence, the capital gains yield of stock X is 0.00 or 00%.

Compute the capital gains yield of Stock Y:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{15}{100}-\frac{20}{100}\\ =0.15-0.20\\ =-0.05\end{array}$

Hence, the capital gains yield of stock Y is -0.05 or -5%.

Compute the capital gains yield of Stock Z:

$\begin{array}{c}\text{Capital gains yield}=\text{Total required return}-\text{Dividend yield}\\ =\frac{15}{100}-\frac{8.8}{100}\\ =0.15-0.088\\ =0.062\end{array}$

Hence, the capital gains yield of stock Z is 0.062 or 6.2%.

Conclusion

Interpretation regarding the relationship among the various returns of each stock:

The entire four stocks have a required rate of return of 15%. However, this return is distributed in a different way between the capital gains and current income. As per the analysis, a higher growth stock has an appreciable capital gains yield but a relatively smaller current income yield.  Moreover, a negative growth stock provides a higher current income even when the price decreases over time.